1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Finding the linear transformation of a matrix

  1. Dec 11, 2009 #1
    1. The problem statement, all variables and given/known data
    Hello again. First of all thanks to anyone who has replied to my previous questions.
    Now, the question that troubles me is:

    We are given a matrix A2x2 with some random values and we are asked to say if there is a linear map which has A as its map for the standar basis.

    A= 2 1
    -5 8

    2. Relevant equations

    3. The attempt at a solution
    If A is the matrix of some linear map for the standar basis that means that:

    f(1,0)=2(1,0) -5(0,1)
    f(0,1)=1(1,0) +8(0,1)

    so f(1,0)=(2,-5) and f(0,1)=(1,8)

    Although the question only asks to say if such a linear map exists (it obviously does but i dont know how to prove it) I would be glad if someone could instruct me on finding formulas for linear maps when we have their matrix.
  2. jcsd
  3. Dec 12, 2009 #2


    Staff: Mentor

    So given a matrix, you are trying to find the linear transformation represented by the matrix. Building on the work you already have, and identifying the transformation by T, we have.

    [tex]T\left(\begin{array}{c}x&y\end{array} \right)[/tex]
    [tex]=~x~T\left(\begin{array}{c}1&0\end{array} \right)~+~y~T\left(\begin{array}{c}0&1\end{array} \right)[/tex]
    [tex]=~x~\left(\begin{array}{c}2&-5\end{array} \right)~+~y~T\left(\begin{array}{c}1&8\end{array} \right)[/tex]
    [tex]=~\left[\begin{array}{cc}?&?&?&?\end{array} \right]~\left(\begin{array}{c}x&y\end{array} \right)[/tex]

    Can you fill in the missing entries in the 2 x 2 matrix?
  4. Dec 12, 2009 #3
    Yes I think I got it now. The missing matrix is the matrix i posted on my first post. So T(x,y)=(2x+y,-5x+8y). The way you put it, I see that a matrix and a linear transformation are actually the exact same thing, expressed in a different way? Thank you very much, the work you people do here is incredible.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook